The correct verb for archery is practice. For example: I am going to practice archery today by shooting some arrows.
It is not archering, archerize, do archery or shooting archery.
You could also say play archery like you would play golf.
Archery is the art, practice, sport or skill of shooting with bow and arrow.
You can use practice or play as a verb.
If you are using a bow and arrow then you are practicing archery as shown in these photos: